I will install in the bulkhead at the feet of the captain's chair on my 2000 X-Star.

Anyone have any insight if one Polk 12" or two 10" MTX subs side-by-side is a better option?

I am not audio-technically inclined. I can plug in wires and feel my way through the install, I just do not know which option will provide better sound output.

I also was told today that I need to make sure the subs do not need a box/enclosure. I assume the MTX doesn't because I had one professionally installed a couple years ago and they did not put it in a box/enslosure. I am not sure about the proposed Polk 12" though.

I personally would skip both of those.....if you want something that will jam in a small enclosure I'd look at a sundown audio SA10, that sub will absolutely get it and really sounds best in .6-.7 cu/ft. For under $200 for the sub they are hard to beat. 600 watts RMS, but you can put more than that to them safely. Stay with a sealed enclosure, it's just easier to build if you don't have alot of experience and make sure you get all the seams sealed with good sealant. Also use birch and not MDF, the MDF will start to break apart with moisture.

gruvn,
MDF (medium density fiberboard) is pressed paper fiber. The Sundown SA-10 woofer cone is pressed paper fiber. It will change its parameters when exposed to high humidity as in the marine environment.
The MTX woofer is a true high 'Q' 'infinite baffle' (IB) woofer that does not require a small and fabricated enclosure. But it still does need an enclosure in the form of complete front to rear acoustic isolation and a very large enclosure which can be created by a helm or seating console. In the case of a true IB 10-inch you would want at least 2.25 cu.ft. up to infinite.
The Polk Audio woofer is 'air suspension' and definitely requires a small fabricated enclosure. It is not intended for infinite baffle. However, you can use it infinite baffle with very limited power handling at very limited output levels. But, not recommended.
Two 10-inch subwoofers have more radiating surface area versus a single 12-inch. In pure output the advantage goes to two 10s if all other things are equal.
Air suspension has more linear excursion under control versus infinite baffle. In pure output the advantage goes to air suspension (small sealed enclosure) if all other things are equal.
So a case can be made for either.
If you are absolutely resolved about "No box", then you must go infinite baffle. A far better IB woofer would be the JL Audio MX 10-inch. These dual 10s would be best matched with an amplifier designed to load into a 2-ohm impedance and deliver an honest 400 watts. If you have more legit power then dual JL Audio M series is a better choice.
If you re-think using a box then I would have other recommendations.
You really cannot select the ideal woofer independent of a matched amplifier and matched enclosure.

I must concur that you are an audio badass. This whole project started because I had to replace my amp. Because I am a student with limited income I needed an inexpensive new amp so I found a Pyle PLMRA400 400-Watt 4-Channel for a good price on Amazon. By installing the new amp I opened the can of worms of toying with the idea of upgrading the sub (to either of the two configurations I previously mentioned). Given that I am committed to this amp now and I already have one 10" MTX, is the idea of running a 2nd 10" MTX off the Pyle worth it or should I just stick to one 10" MTX with the Pyle?

Sub setups are always a world of compromise. Your goals and limitations should be your start point. I used to compete in SPL competition and everything was tuned to a specific frequency with virtually no size and power constraints. I used two Rockford HX2 woofers paralleled to 1 Ohm on a crossfire 1000D in a ported box tuned closely to the resonate frequency of the explorer. Those 2 10s killed within a band of about 4Hz. Any other frequency left plenty to be desired in sound quality. Sound quality and wide frequency response require a sealed enclosure. Free air woofers will never quite give you the precise punch of a sealed enclosure, but will provide some lows with minimal space and work requirements. This all brings me back to the point of what are your constraints and desired output? If you want clear precise low tones over a broader spectrum but not defining bass consider building an enclosure. If you have no room, or time choose a free air type. If you want to test every lock washer in the boat get ready to fork over serious space and design a box that is tuned to a low frequency. There are plenty of woofers in the 10 range that can deliver punch in a small enclosure. Look into dual voice coil woofers if you have the power (and 2 Ohm stable) amp. A good DVC 10 will out preform 2 cheap 10s, or a single 12 if properly setup, I am not sure if there are any good DVC subs in the marine market though. I would have to research that.
Good luck!

I must concur that you are an audio badass. This whole project started because I had to replace my amp. Because I am a student with limited income I needed an inexpensive new amp so I found a Pyle PLMRA400 400-Watt 4-Channel for a good price on Amazon. By installing the new amp I opened the can of worms of toying with the idea of upgrading the sub (to either of the two configurations I previously mentioned). Given that I am committed to this amp now and I already have one 10" MTX, is the idea of running a 2nd 10" MTX off the Pyle worth it or should I just stick to one 10" MTX with the Pyle?

The lure of a low price and inflated specifications didn't get you a useful amplifier. The Pyle model referenced is a BTL amplifier with no power supply, no crossover and cannot be bridged. It will struggle with a 2-ohm load. Current consumption is 5 amps. In reality this amplifier has no more power than a standard 4-channel head unit which is about 4 times 16 watts. This amplifier is good for driving four entry level coaxials when at rest if you are not having to compete with noise when underway.
Don't let those limited dollars burn a hole in your pocket. Wait and save so that you can afford something of lasting value.

The lure of a low price and inflated specifications didn't get you a useful amplifier. The Pyle model referenced is a BTL amplifier with no power supply, no crossover and cannot be bridged. It will struggle with a 2-ohm load. Current consumption is 5 amps. In reality this amplifier has no more power than a standard 4-channel head unit which is about 4 times 16 watts. This amplifier is good for driving four entry level coaxials when at rest if you are not having to compete with noise when underway.
Don't let those limited dollars burn a hole in your pocket. Wait and save so that you can afford something of lasting value.